title: “Statistical methods for movement ecology” author: Marie-Pierre Etienne institute: "" date: “2022” csl: “../resources/apa-no-doi-no-issue.csl” output: xaringan::moon_reader: css: [‘metropolis’, ‘mpe_pres_HDR.css’] lib_dir: libs nature: ratio: 16:10 highlightStyle: github highlightLines: true countIncrementalSlides: false beforeInit: ‘../courses_tools/resources/collapseoutput.js’ — name: intro

Movement ecology paradigm

(Nathan, Getz, Revilla, et al., 2008) presents individual movement as the results of

Motion capacities
Internal state
Environment ]

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Time series with values in \(\mathbb{R}^2\) (on earth …).

\[\begin{array}\\ \mbox{Time} & \mbox{Location} & \mbox{Turning angle} & \mbox{Speed}\\ t_{0} & (x_0, y_0) & NA & NA\\ t_{1} & (x_1, y_1) & NA & sp_1\\ t_{2} & (x_2, y_2) & ang_2 & sp_2\\ \vdots & \vdots& \vdots& \vdots \\ t_{n} & (x_n, y_n) & ang_n & sp_n\\\\ \end{array}\] ]

Ecological questions

Behavioural ecology

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Ecological questions

Behavioural ecology

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Habitat preference

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Often analysed using discrete time model (McClintock, Johnson, Hooten, et al., 2014)

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Segmentation

Heterogeneity in movement pattern interpretated as different internal states

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Classically addressed with Hidden Markov Model

Exploring the change point detection approach. (Lavielle, 2005; Picard, Robin, Lebarbier, et al., 2007)

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Segmentation

Signal processing approach for movement ecology (Picard, Robin, Lebarbier, et al., 2007)

Let \(\boldsymbol{\tau}={\tau_1,...,\tau_{K-1}}\) ( \(\tau_0=-1\) ) be a partition in K segments of \(\{1,\ldots n\}\) \[\boldsymbol{X}_{i}\overset{i.i.d}{\sim}\mathcal{L}(\theta_k),\quad \forall i \in \{ \tau_{k-1}+1:\tau_k \}\] The .care[Dynamic Programming algorithm] allows to explore efficiently all possible segmentation and to estimate \(\boldsymbol{\hat{\tau}}\) ]

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Let \(Z_k\) stand for the class of segment \(k,\) \(\forall i \in \{ \tau_{k-1}+1:\tau_k \}\) \[Z_k \overset{i.i.d}{\sim} \mathcal{M}(\pi), \quad\boldsymbol{X}_{i}\vert Z_k=l \overset{i.i.d}{\sim}\mathcal{L}(\theta_l)\] The Dynamic Programming coupled with EM algorithm allows to explore efficiently all possible segmentation and to estimate \(\boldsymbol{\hat{\tau}}\) ]

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In (Patin, Etienne, Lebarbier, et al., 2019) : a direct extension to simultaneous segmentation for home range shift.

Segmentation

Signal processing approach for movement ecology (Lavielle, 2005; Picard, Robin, Lebarbier, et al., 2007)

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Pros and cons of Discrete time versus continuous time movement models discussed in McClintock, Johnson, Hooten, et al. (2014)

Potential based Diffusions as continuous time movement model

Let \((X_s)_{s\geq0}\in\mathbb{R}^2\) denote the position at time \(s\).

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– .pull-left[ * Ornstein Uhlenbeck process: central place behavior \[dX_s = -B (X_s- \mu) ds + dW_s, \quad X_0=x_0.\]

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Popular models as Brownian Motion and Ornstein Uhlenbeck have known transition densities \(q(x_t, x_{t+s})\) which is not the case in general.

Potential based Diffusions as continuous time movement model

Brillinger, Preisler, Ager, et al. (2002) propose a flexible framework \[dX_s = -\nabla H(X_s) ds + \gamma dW_s, \quad X_0=x_0.\]

but no explicit transitions \(q(x_t, x_{t+s})\)

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In Gloaguen, Etienne, and Le Corff (2018a), as part of P. Gloaguen’s PhD, explore \(H(X_s) = \sum_{k=1}^K \pi_k \varphi_k(X_s),\)

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.pull-right[ * Euler approximation : biased estimates with low frequency data * (Ozaki, 1992) and Kessler (1997) same results than * MCEM based on exact simulation (Beskos, Papaspiliopoulos, Roberts, et al., 2006) limits the flexibility of the SDE.]

Potential based Diffusions as continuous time movement model

Partially observed SDE

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.pull-rights[ Let \(Y_k\) be the recorded position \(s_k\), a noisy observation of the true position \(X_k\):

\[dX_s = b(X_s) ds + \gamma dW_s, \quad X_0=x_0; \quad Y_k \overset{ind}{\sim} \mathcal{L} (X_k, \theta{o}).\] ]

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Additive smoothing distributions for the E Step

\[\sum_{k=0}^{n-1}\mathbb{E}( h(X_k, X_{k+1}) \vert Y_{0:n})\]

* The particle-based, rapid incremental smoother (PaRIS) algorithm (Olsson and Westerborn, 2017) provides an online smoother using a rewriting of the Backward weight and an acceptation/rejection mechanism but depends on \(q(\xi_{k-1}, \xi_{k})\)

* The generalized random PaRIS algorithm, in (Gloaguen, Etienne, and Le Corff, 2018b), uses simple Euler approximation to propose the particles and uses a General Poisson Estimator to replace \(q(\xi_{k-1}, \xi_{k})\) with an unbiased estimator.

.care[Restrictive constraints on the drift and the diffusion term], are relaxed in (Martin, Etienne, Gloaguen, et al., 2021).

Potential based Diffusions as continuous time movement model

Flexible movement model for habitat preference

Potential based Diffusions as continuous time movement model

Flexible movement model for habitat preference

Potential based Diffusions as continuous time movement model

Flexible movement model for habitat preference

A classical choice of resource selection function, (i.e stationary distribution including covariates) \[\pi{\left(x \vert \beta\right)} \varpropto \exp\left(\sum_{j=1}^J \beta_j c_j (x) \right). \]

Diffusion, under regularity condition admits a stationary distribution.

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Combining the ideas of (Michelot, Blackwell, and Matthiopoulos, 2019), and (Brillinger, Preisler, Ager, et al., 2002) lead to the Langevin diffusion as movement model, \[ d X_t = \frac{\gamma^2}{2} \nabla \log \pi{\left(X_t\right)} \, d t + \gamma \,d W_t,\quad X_0 =x_0. \] –

Using Euler approximation

\[ X_{i+1} \vert \lbrace X_i = x_i \rbrace = x_i + \frac{\gamma^2 \Delta_i}{2} \sum_{j=1}^J \beta_j \nabla c_j(x_i) + \sqrt{\Delta_i} \varepsilon_{i+1},\quad \varepsilon_{i+1} \overset{ind}{\sim} N \left( {0} , \gamma^2 \boldsymbol{I}_d \right),\]

leads to a simple linear model published in Michelot, Etienne, Blackwell, et al. (2019).

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Potential based Diffusions as continuous time movement model

Flexible movement model for habitat preference

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Perspectives

Some natural extension of the Langevin model useful in movement ecology

Coupling Hidden Markov model or change point detection model with a Langevin distribution:
Introduction of an individual random effect,

Handling categorical covariates

Lejay and Pigato (2018) define a threshold diffusion and proposes an ML estimation method. Explore the generalisation to \(\mathbb{R}^2\).

Longer term perspective

Collective movement: Cheaper GPS imply massive deployment. Invest the area of collective movement analysis.
Combined sound monitoring

# A few words to finish

* Close interaction with biologist, - it’s helpful, - provides exciting statistical problems, - experiment a large diversity of approaches. * Hidden variables approach - provide flexible models (spatial abundance, classification in time series, Partially observed SDE) - popularized in Ecology in a Bayesian setting but frequentist approach is also possible * The Markovian property - not so realistic, - but a key component in both theoretical approach and computational aspects

Beskos, A., O. Papaspiliopoulos, G. O. Roberts, et al. (2006). “Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion)”. In: Journal of the Royal Statistical Society: Series B (Statistical Methodology) 68.3, pp. 333-382.

Brillinger, D. R., H. K. Preisler, A. A. Ager, et al. (2002). “Employing stochastic differential equations to model wildlife motion”. In: Bulletin of the Brazilian Mathematical Society 33.3, pp. 385-408.

Gloaguen, P., M. Etienne, and S. Le Corff (2018b). “Online sequential Monte Carlo smoother for partially observed diffusion processes”. In: EURASIP Journal on Advances in Signal Processing 2018.1, p. 9.

Gloaguen, P., M. Etienne, and S. Le Corff (2018a). “Stochastic differential equation based on a multimodal potential to model movement data in ecology”. In: Journal of the Royal Statistical Society: Series C (Applied Statistics) 67.3, pp. 599-619. DOI: 10.1111/rssc.12251.

Kessler, M. (1997). “Estimation of an ergodic diffusion from discrete observations”. In: Scandinavian Journal of Statistics 24.2, pp. 211-229.

Lavielle, M. (2005). “Using penalized contrasts for the change-point problem”. In: Signal processing 85.8, pp. 1501-1510.

Lejay, A. and P. Pigato (2018). “Maximum likelihood drift estimation for a threshold diffusion”. In: Scandinavian Journal of Statistics.

Martin, A., M. Etienne, P. Gloaguen, et al. (2021). “Backward importance sampling for online estimation of state space models”. working paper or preprint. URL: https://hal.archives-ouvertes.fr/hal-02476102.

McClintock, B. T., D. S. Johnson, M. B. Hooten, et al. (2014). “When to be discrete: the importance of time formulation in understanding animal movement”. In: Movement Ecology 2.1, p. 21.

Michelot, T., P. G. Blackwell, and J. Matthiopoulos (2019). “Linking resource selection and step selection models for habitat preferences in animals”. In: Ecology 100.1.

Michelot, T., M. Etienne, P. Blackwell, et al. (2019). “The Langevin diffusion as a continuous-time model of animal movement and habitat selection”. In: Methods in Ecology and Evolution.

Nathan, R., W. M. Getz, E. Revilla, et al. (2008). “A movement ecology paradigm for unifying organismal movement research”. In: Proceedings of the National Academy of Sciences 105.49, pp. 19052-19059.

Olsson, J. and J. Westerborn (2017). “Efficient particle-based online smoothing in general hidden Markov models: the PaRIS algorithm”. In: Bernoulli 23.3, pp. 1951-1996.

Ozaki, T. (1992). “A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach”. In: Statistica Sinica, pp. 113-135.

Patin, R., M. Etienne, E. Lebarbier, et al. (2019). “Identifying stationary phases in multivariate time series for highlighting behavioural modes and home range settlements”. In: Journal of Animal Ecology.

Picard, F., S. Robin, E. Lebarbier, et al. (2007). “A segmentation/clustering model for the analysis of array CGH data”. In: Biometrics 63.3, pp. 758-766.